OFFSET
0,3
COMMENTS
Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.
LINKS
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
FORMULA
Conjecture: a(n) = A052161(n - 2) + 1.
EXAMPLE
The a(4) = 35 edge-sets:
{} {12} {12,13} {12,13,14} {12,13,14,34}
{13} {12,14} {12,13,23} {12,13,23,34}
{14} {12,23} {12,13,34} {12,14,24,34}
{23} {12,24} {12,14,24} {12,23,24,34}
{24} {13,14} {12,14,34}
{34} {13,23} {12,23,24}
{13,34} {12,23,34}
{14,24} {12,24,34}
{14,34} {13,14,34}
{23,24} {13,23,34}
{23,34} {14,24,34}
{24,34} {23,24,34}
MATHEMATICA
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Length[csm[#]]<=1&&!MatchQ[#, {___, {x_, y_}, ___, {z_, t_}, ___}/; x<z<y<t||z<x<t<y||x<z<t<y||z<x<y<t]&]], {n, 0, 5}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 29 2019
STATUS
approved